Chapter 10Solutions: NumericalMethods for ConstrainedOptimizationCHALLENGE 10.1.The graphical solutions are left to the reader.(a) The optimality condition is that the gradient should be zero. We calculateg(x)=·2x1−x2+58x2−x1+3¸,H(x·2−1−18¸.SinceHis positive deFnite (Gerschgorin theorem),f(x) has a unique minimizersatisfyingg(x0,soH(x)x=·−5−3¸,and thereforex=[−2.8667,−0.7333]T.(b) Using a Lagrange multiplier we obtainL(x,λx21+4x22−x1x2x1x2+6−λ(x1+x2−2).
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